Computing Signatures for Representations of the Hecke and Cherednik Algebras UROP+ Final Paper, Summer 2015 Saarik Kalia Mentor: Siddharth Venkatesh Project suggested by Pavel Etingof September 1, 2015 Abstract In this paper, we compute the signatures of the contravariant form on Specht modules for the cyclotomic Hecke algebra and compute the signature character of the contravariant form on the polynomial representation of the rational Cherenik algebra associated to G(r, 1, n).

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1

Introduction

Let A be an algebra and V a left A-module which admits a non-degenerate invariant Hermitian form. The problem of determining whether this form is positive-definite is an important one in representation theory. This problem has been explored for the rational Cherednik algebra by Etingof and Stoica in [ES], and for the cycolotomic Hecke algebra by Stoica in [S]. In this paper, we consider the more general problem of determining the signature of this form. We define the signature of a form on a finite-dimensional vector space as follows. Definition. Let V be a finite-dimensional vector space with non-degenerate Hermitian form h·, ·i. Let {ei } be a basis for V which is orthogonal with respect to this form. Then the signature s(V ) of V is the number of basis elements with positive norm minus the number of basis elements with negative norm. Now for an infinite-dimensional vector space with a natural grading, we may define the signature character as follows. Definition. Let V be a vector space withL non-degenerate Hermitian form h·, ·i. Suppose there exists a grading V = ∞ m=0 Vm so that Vm and Vn are orthogonal when m 6= n. Then we may define the signature character chs (V ) =

∞ X

tw s(Vw )

w=0

of V with respect to this form. In Section 2, we present the definition of the cyclotomic Hecke algebra, as well as some preliminary theorems that will state what its irreducible representations are and how to compute in them. In Section 3, we derive a formula for the signature of all the irreducible representations of the Hecke algebra. In Section 4, we present definitions and preliminaries for the rational Cherednik algebra and its polynomial representation. Finally in Section 5, we compute the signature character of this representation.

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Preliminaries for the Hecke Algebra

Before we introduce the Hecke algebra, we first motivate its definition by introducing complex reflection groups. 2

Definition. Let h be a finite dimensional vector space over C. A reflection of h is a unitary transformation s of h with rk(s − 1)|h = 1. If a finite group W is generated by reflections of h, we say that W is a complex reflection group acting on h. Throughout this paper, we will be dealing with the complex reflection group G(m, 1, n) = Sn n (Z/mZ)n . This group can also be expressed with the following generators and relations. Theorem 1 ([AK], Proposition 2.1). The complex reflection group G(m, 1, n) is generated by s0 , s1 , . . . , sn−1 subject to sr0 = 1 s2i = 1 for i > 0 si sj = sj si if |i − j| > 1 si si+1 si = si+1 si si+1 for i > 0 s0 s1 s0 s1 = s1 s0 s1 s0 . This motivates the definition of the cyclotomic Hecke algebra as a deformation of the above complex reflection group. Definition. Let R be a commutative domain, with k its field of fractions, and take q, q1 , . . . , qm ∈ R× . The cyclotomic Hecke algebra HR,n = HR,n (q, q1 , . . . , qm ) is defined as the unital associative R-algebra generated by the elements T0 , . . . , Tn−1 subject to (T0 − q1 ) · · · (T0 − qm ) = 0 (Ti − q)(Ti + 1) = 0 for i > 0 Ti Tj = Tj Ti if |i − j| > 1 Ti Ti+1 Ti = Ti+1 Ti Ti+1 for i > 0 T0 T1 T0 T1 = T1 T0 T1 T0 . Throughout this paper, we take R = C and we take q, q1 , . . . , qm to have norm 1 and be generic. (In particular, we require that qqji q a 6= 1 for any i, j, a.) We first introduce the notion of m-partitions and m-tableaux because the irreducible representations of the Hecke algebra will be given by its actions on the standard Young m-tableaux. 3

Definition. An m-partition λ = (λ1 , . . . , λm ) of n is ordered m-tuple so Pan m i i that each λ is a partition of an integer |λ |, with i=1 |λi | = n. We then write λ `m n. Definition. Let λ = (λ1 , . . . , λm ) `m n. The Young m-diagram [λ] of shape λ is the m-tuple of Young diagrams ([λ1 ], . . . , [λm ]). We call [λi ] the components of [λ]. A standard Young m-tableau of shape λ is an enumeration from 1 to n of the boxes of [λ] so that each row and each column of each component is increasing. We denote the set of all standard Young m-tableaux of shape λ by Std(λ) and its formal k-linear span by Vλ . Finally, we define some notation which will be necessary to state our main preliminary theorem. Definition. Let λ `m n and s ∈ Std(λ). For 1 ≤ i ≤ n, the content cs (i) of i in s is the row index of i in its component of s minus the column index of i in its component of s. For 1 ≤ a, b ≤ n, we define the axial distance rs (a, b) = cs (a) − cs (b). We also denote the index of the component of i in s by τs (i). Theorem 2 ([AK], Theorems 3.7 and 3.10). HR,n is semisimple. Its irreducible representations are exactly Vλ for all λ `m n, with the action of Ti on s ∈ Std(λ) given by: • T0 s = qτs (1) s. • For i > 0, if i and i + 1 lie in the same row of the same component of s, then Ti s = qs. • For i > 0, if i and i + 1 lie in the same column of the same component of s, then Ti s = −s. • For i > 0, if neither of the above hold, let t = (i, i + 1)s be the standard m-tableau gotten by swapping the positions of i and i + 1 in s. Then q− 1−q Ti s = − s + q s (i) r(i+1,i) 1 − qτ τ(i+1) q 1− s

qτs (i) r(i+1,i) q qτs (i+1) qτs (i) r(i+1,i) t q qτs (i+1)

Moreover, [S] (Proposition 3.2) defines a non-degenerate Hermitian form h·, ·i on Vλ which is HR,n -invariant (i.e. hv, wi = hTi v, Ti wi for all 0 ≤ i ≤ n − 1 and v, w ∈ Vλ ), and shows that s ∈ Std(λ) form an orthogonal basis with respect to h·, ·i. This will be the form with respect to which we compute the signature. 4

3

Signature for the Representations of the Hecke Algebra

Now we move on to compute the signature of the representation. First we derive a formula for how the norm of an element changes when we switch its entries. Proposition 1. Let s, t be standard m-tableaux of shape λ, with t = (i, i+1)s. Then # " 1 ht, ti = 2 · Re q s (i) r (i+1,i) hs, si. qs q − qτ τ(i+1) s

Proof. For convenience, write a = τs (i), b = τs (i + 1), and r = rs (i + 1, i). We have q − qqab q r 1−q s+ t. Ti s = − 1 − qqab q r 1 − qqab q r Since h·, ·i is HR,n -invariant, then 1 − q 2 q − hs, si = hTi s, Ti si = qa r hs, si + 1 − qb q 1 −

qa r 2 q qb qa r q qb



ht, ti.

Rearranging, we get 2 qa r 1 − qb q − |1 − q|2 q + q¯ − qqab q r − qqab q r hs, si = hs, si ht, ti = 2 2 q − qqab q r q − qqab q r ! " # 1 1 1 = + = 2 · Re hs, si. q − qqab q r q − qqa q r q − qqab q r b

We now establish a distinguished m-tableau whose norm we will use to compute the norms of all other m-tableaux. In addition we define some notation which will important for stating our result. Definition. Consider the standard m-tableau t0 ∈ Std(λ) gotten by putting the numbers λi−1 + 1, . . . , λi in the ith component (where λ0 = 0), and arranging the numbers within each component in consecutive increasing order across rows. 5

Definition. For s ∈ Std(λ) and 1 ≤ i ≤ n, let js (i) denote the number lying in the box of t0 corresponding to the box of s in which i lies. Then we say a and b are inverted in s if (a − b)(js (a) − js (b)) < 0, and we write a ↔ b. s

Definition. For z ∈ C× , let {z} =

Re z . |Re z|

Now we prove the general signature formula for irreducible representations of the Hecke algebra. Theorem 3. Taking the convention that ht0 , t0 i > 0, we have  X Y  qτs (a) rs (b,a) s(Vλ ) = q− q . qτs (b) 1≤a i | µj ≤ µi }| + 1. P ij Definition. Let di = m−1 j=1 ζm cj . Definition. Let αa,b = aκ − brc0 . Theorem 4 ([G1], Theorem 6.1 and Corollary 6.2). There exist operators σi , Φ, Ψ ∈ Hκ,c for 1 ≤ i ≤ n − 1 and polynomials fµ = xµ + o(xµ ) ∈ V for all µ ∈ Zn≥0 which satisfy   fsi µ , if µi 6= µi+1 (mod r) or µi < µi+1 0, if µi = µi+1 σi fµ =  Cfsi µ , if µi = µi+1 (mod r) and µi > µi+1 , where C=

ακ(µi −µi+1 ),vµ (i)−vµ (i+1)−1 · ακ(µi −µi+1 ),vµ (i)−vµ (i+1)+1 , (ακ(µi −µi+1 ),vµ (i)−vµ (i+1) )2 Φfµ = fφµ ,

and

 Ψfµ =

0, if µn = 0 (αµn ,vµ (n)−1 − d0 + d−µn )fψµ , if µn 6= 0.

Here si acts on an n-tuple by exchanging the ith and (i + 1)th entries, φ acts by φ(µ1 , . . . , µn ) = (µ2 , . . . , µn , µ1 + 1), and ψ acts by the inverse of φ. Moreover in [G2] (Section 6), it is shown that the fµ ’s form an orthogonal basis with respect to h·, ·i, the σi ’s are self-adjoint, and Φ is adjoint to Ψ. This will allow us to compute the signature character of V . 8

5

Signature Character for the Polynomial Representation

Now we present the formula for the norm of fµ , and as a corollary we derive the formula for the signature character of V . Theorem 5. Let µ ∈ Zn≥0 . Fix some nondecreasing reordering of the entries of µ. Let g(i) denote the index of µi in this reordering. Also let p(i, x) denote the number of entries of µ which are greater than µi + x or which are equal to µi + x and have index less than i. Then ! ! ∞ µi n n Y Y Y αjm,n Y αjm,n−g(i)−p(i,jm) . hfµ , fµ i = (αj,g(i)−1 − d0 + d−j ) · α α jm,0 jm,n−g(i)−p(i,jm)+1 i=1 i=1 j=1 j=1 Proof. Because the operators σi are self-adjoint, then when λi 6≡ λi+1 (mod m), we have hfλ , fλ i = hσi fsi λ , fλ i = hfsi λ , σi fλ i = hfsi λ , fsi λ i, and when λi ≡ λi+1 (mod m) with λi > λi+1 hfλ , fλ i = hσi fsi λ , fλ i = hfsi λ , σi fλ i αλ −λ ,v (i)−vλ (i+1)+1 · αλi −λi+1 ,vλ (i)−vλ (i+1)−1 hfsi λ , fsi λ i. = i i+1 λ (αλi −λi+1 ,vλ (i)−vλ (i+1) )2 Likewise since Φ and Ψ are adjoint, we have (for λn 6= 0) hfλ , fλ i = hΦfψλ , fλ i = hfψλ , Ψfλ i = (αλn ,vλ (n)−1 − d0 + d−λn )hfψλ , fψλ i. Now we describe a sequence of si and φ operations which will lead us from the zero string 0n = (0, . . . , 0) to µ, and we will calculate the norm of fµ by multiplying the factors we acquire when traversing the same sequence with σi and Φ operations acting on f0n . We begin by applying φn to 0n , and repeat this µh(1) times so that we are left with the string µnh(1) = (µh(1) , . . . , µh(1) ) (where h denotes the inverse of g). Now we apply φn−1 and then sn−1 sn−2 . . . s1 so that we are left with the string (µh(1) + 1, . . . , µh(1) + 1, µh(1) ). We repeat this µh(2) − µh(1) times to leave us with (µh(2) , . . . , µh(2) , µh(1) ). Next we will apply φn−2 followed by sn−2 . . . s1 sn−1 . . . s2 , and repeat this µh(3) −µh(2) times to get the string (µh(3) , . . . , µh(3) , µh(2) , µh(1) ). We continue in this way, incrementing n − i of the entries and passing the 9

other i entries back to the front so that they are not incremented, until the value of the n − i entries is µh(i) , at which point we repeat for i + 1. The one exception is that for each pair i, j such that µi ≡ µj (mod m) with µi > µj and i > j, we do not pass one entry of value µj through one entry of value µi . Thus at the end of the process we have a string with all the entries of µ, so that all entries with the same value modulo m appear in the string in the same order as they appear in µ. We may then freely reorder the entries using the si operations, without switching any entries µi ≡ µj (mod m), to arrive at the string µ. The corresponding actions of σi will therefore contribute no factors, so this final step does not change the norm. Now we first analyze the factors acquiredQ from the actions of Φ. The first n times we apply it, we acquire the factors ni=1 (α1,i−1 − d0 + d−1 ) because λn = 1 (after the action) and while the first time, all j − 1 other entries in λ are less than 1 (after the action), each successive time, one less entry is strictly less than Likewise the j th time we increment the entries, the factors Q1. we acquire are ni=n−k+1 (αj,i−1 − d0 + d−j ), where k is the number of entries we increment. Note that the number of times i − 1 appears as the second argument of α is µh(i) (in particular it appears for j = 1, . . . , µh(i) ). Replacing the indexing variable i by g(i), we find that the total factor acquired is µi n Y Y (αj,g(i)−1 − d0 + d−j ). i=1 j=1

Now we analyze the factors acquired from the actions of σi . Unless λi ≡ λi+1 (mod m), we acquire no factor. Therefore we only acquire factors when we pass an entry of value µi to the front at the (µi +jm)th step. In particular, we acquire a factor for each (i, j, k) such that µk − µi ≥ jm (equality only contributes a factor if k < i). For fixed i, j, the factors we acquire are p(i,jm)

Y αjm,n−g(i)−k αjm,n−g(i)−k+2 αjm,n−g(i)−p(i,jm) αjm,n−g(i)+1 = . 2 (α α jm,n−g(i)−k+1 ) jm,n−g(i)−p(i,jm)+1 αjm,n−g(i) k=1

Taking the product over all i, j we get ∞ Y n Y αjm,n−g(i)−p(i,jm) αjm,n−g(i)+1 j=1 i=1

αjm,n−g(i)−p(i,jm)+1 αjm,n−g(i)

∞ n Y αjm,n Y αjm,n−g(i)−p(i,jm) = . α α jm,0 jm,n−g(i)−p(i,jm)+1 j=1 i=1

Combining these two factors and taking hf0n , f0n i = 1 proves the formula. (Note that if µk − µi < jr for all i, k, then p(i, jm) = 0 for all i, and the j th 10

term in this product will cancel out to leave 1. Since this is true for all j sufficiently large, then all but finitely many of the terms in this product are in fact 1.) Corollary 1. The signature character of the polynomial representation is ( n µ ) n ∞ ∞ i Y Y X Y Y X α α jm,n−g(i)−p(i,jm) jm,n (αj,g(i)−1 − d0 + d−j ) · . tw α α jm,0 jm,n−g(i)−p(i,jm)+1 n w=0 j=1 i=1 j=1 i=1 µ∈Z ≥0

|µ|=w

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Conclusion and Future Research

In this paper, we employed computational results from other works to calculate the signature of all finite-dimensional irreducible representations of the cyclotomic Hecke algebra and the signature character of the polynomial representation of the rational Cherednik algebra. As future research, we would certainly like to calculate the signature character of all irreducible representations of the rational Cherednik algebra, for which there already exist similar computational tools. We would also like to see how our results behave when we take assymptotic limits of the parameters. Finally, we would consider exploring how we might be able to undergo this Cherednik algebra calculation in a basis which is preserved by the action of Sn (unlike the fµ ’s that we used).

Acknowledgments I would like to thank Siddharth Venkatesh for all he has taught me, and for all the help and direction he has given me on this project. I would also like to thank Professor Etingof for suggesting this project and answering our questions along the way. Finally, I would like to thank the MIT Math department and the UROP+ program for allowing us to perform this research.

References [AK] S. Ariki, K. Koike, A Hecke algebra of Z/rZ o Sn and construction of its irreducible representations, Adv. Math 106 (1994), pp. 216-243. 11

[ES] P. Etingof, E. Stoica, Unitary repreeesentations of rational Cherednik algebras (with an appendix by Stephen Griffeth), Representation Theory 13 (2009), pp. 349-370. [G1] S. Griffeth, Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n)., arXiv:math/0612733. [G2] S. Griffeth, Orthogonal functions generalizing Jack polynomials, arXiv:math/0707.0251 [S] E. Stoica, Unitary representations of Hecke algebras of complex reflection groups, arXiv:math/0910.0680. [V] V. Venkateswaran, Signatures of representations of Hecke algebras and rational Cherednik algebras, arXiv:math/1409.6663

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